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Chapter 8
X-ray spectrum
X-ray was discovered by W. K. Röntgen in 1895 in a
cathode tube, weak fluorescence on screen (BaPt(CN)6
铂氰酸钡)
In history, X-ray is called
for its mysterious and
unknown properties.
The X-ray spectra led to
the theory of the shell
structure of the
atom (Kossel 1914)
Wilhelm Conrad Roentgen (1845–1923).
(Courtesy of AIP Emilio Segré Visual Archives)
By “X-rays”, we usually mean electromagnetic radiation (light)
which has a wavelength shorter than that of ultraviolet light —
though there is no sharp boundary. The range is usually
considered to be 0.1 to 10 , which corresponds to quantum
energies of 1 — 100keV.
 X-ray imaging of bones,
 X-ray diffraction in crystal,
 Compton effect,
 Expose films,
X-ray generations
X-rays are usually generated by irradiating an anode (anti-cathode)
with fast electrons.
They may also be produced by electron impact or collisional
excitation of free atoms, and thus independently of solid state
influences.
Early X-Ray Tube (1900):
This tube was used for
diagnostic radiography and
was made about 1900.
Early X-Ray Tube (1904): Used for
radiation treatment of skin cancers,
was made about 1904. An interesting
feature
is
the
glass
'cone',
incorporated
to
allow
easy
reproduction of treatment distances
and alignments
X-ray spectra
If an anti-cathode is bombarded with electrons which
have passed through an accelerating voltage V0, x-rays are
generated. Spectral analysis of these reveals that:
 There is always a continuum spectrum, the x-ray
bremsstrahlung;
 And under certain conditions, there is in addition a line
spectrum, the characteristic spectrum.
X-ray bremsstrahlung
"Bremsstrahlung" means "braking radiation" and is retained from the original German to
describe the radiation which is emitted when electrons are decelerated or "braked" when
they are fired at a metal target. Accelerated charges give off electromagnetic radiation,
and when the energy of the bombarding electrons is high enough, that radiation is in the
x-ray region of the electromagnetic spectrum. It is characterized by a continuous
distribution of radiation which becomes more intense and shifts toward higher
frequencies when the energy of the bombarding electrons is increased.
The lines are superimposed on the
bremsstrahlung spectrum.
Characteristic x-rays are emitted
from heavy elements when their
electrons make transitions
between the lower atomic energy
levels.
The characteristic x-rays emission which shown as two sharp peaks in the
illustration at left occur when vacancies are produced in the n=1 or K-shell of the
atom and electrons drop down from above to fill the gap. The x-rays produced by
transitions from the n=2 to n=1 levels are called K-alpha x-rays, and those for the
n=3->1 transiton are called K-beta x-rays.
Explanation for the bremsstrahlung spectrum
If the intensity is plotted against the frequency, the
bremsstrahlung spectrum for an accelerating voltage V0 is
described by
I ( )  const  Z ( max  )
Where I is the intensity of the radiation (energy per time and
frequency interval and solid angle), and Z is the atomic number
of the anticathode material. The limiting frequency is given by:
h max  e V0
This means that the high-energy or short-wavelength limit of the
x-ray spectrum max is given by the energy equivalent eV0.
The bremsstrahlung spectrum is a result of the fact that when
electrons pass through close to the atomic nuclei, they are
deflected and slowed down. A positive or negative accelerated
charge will, according to classical electrodynamics, emit
electromagnetic radiation. This is “white” or continuous x-ray
bremstrahlung.
In terms of quantum theory, this can be understood as follows:
For each braking incident, a quantum of light h = E0 – E is
emitted. However, since the beginning and end states are not
quantised – the electrons are free, not bound – a “white”
spectrum arises when there are many individual events.
The reaction equation is:
Atom + e- (fast)  Atom + e- (slow) + h
h
e-, E0
+Ze
e-, E < E0
In the limiting case, the entire energy of the electron is
emitted in a single quantum in the course of a single
braking event.
Explanation for the characteristic spectrum
Characteristic radiation consists of a relatively small number of
lines. The lines are again grouped into series, which converge to
a short-wavelength limit, which is called an “edge”. In general it
holds for characteristic spectra that x-ray spectra include a
limited number of lines which can be grouped into a few series.
With a rhodium anticathode, for example, one can observe the
following lines and series by increasing the accelerating voltage
on the electrons in steps:
For accelerating voltages V0 > 0.5kV, the lines of the M series,
For accelerating voltages V0 > 3.0kV, the L series also,
For accelerating voltages V0 > 23kV, the K series as well,
The lines of the K series are doublets.
Spectral positions of the
characteristic x-ray emission
lines and the absorption edges of
the elements. The quantum
energies increase with increasing
nuclear charge number.
There is a clear relationship to the nuclear charge. Corresponding lines and
edges are found at increasing quantum energies as the nuclear charge
increases. The series are designated by the letters K, L, M, N, … and the lines
within the series by Greek lower case letters beginning with . The fine
structure splitting of the lines is indicated by numbers written as subscripts.
To a good approximation, the first line of the K series, the line
K, can be described for atoms with different nuclear charge
numbers Z by the expression:
1
2
2 1
3
 K  4 R( Z  1)  R( Z  1) ( 2  2 )
1 2
The first lines of the L series (L) are described by:
 L 
5
36
1 1
R( Z  7.4)  R( Z  7.4) ( 2  2 )
2 3
2
2
A linear relationship between 1/2 and the nuclear charge number
Z for analogous x-ray lines or edges in the spectra of different
elements was discovered in 1913 by Moseley. Comparing with
Balmer formula for hydrogen suggests that for the K line the
nuclear charge is screened by one unit of charge, while for the L
line, it is screened by almost eight units.
Moseley plot
Chemical bonding of an atom has only a slight influence on
its x-ray spectrum. However, exact measurement of this effect
does provide important information about the behavior of
electrons in chemical bonds. This is importance in molecular
and solid-state physics.
The emission of x-rays can be elicited not only by
bombarding an anticathode with electron, but also by
irradiation of atoms, molecules or solids with x-rays. This is
called x-ray fluorescence.
The wavelength of the x-radiation is greater than, or at least equal to, that of
the exciting light, but other than that, it is independent of the wavelength of
the exciting radiation within certain limits. The lines of a series appear in a
fluorescence spectrum, and then all of them at once, only when the quantum
energy of the exciting radiation is at least as great as the quantum energy of
the highest-energy, or shortest-wavelength line in the characteristic spectrum.
It is the same with excitation of x-radiation by electron bombardment: the
kinetic energy of the electrons eV0 must be at least as great as the quantum
energy of the shortest-wavelength line of the series before this series appears
in the emission spectrum. Thus emission of the K line cannot be excited by
the quantum energy of K; instead it is necessary to supply the energy of the
K edge. This is the energy which the lines of the K series converge, the
series limit.
From this and other observations, it was concluded that x-ray lines
correspond to states of “inner” electrons which are bound in filled shells, in
contrast to the more loosely bound outer electrons, which give rise to the
optical spectra.
Kossel’s interpretation
In 1916, Kossel interpreted the generation of the x-ray line
spectra as follows: first the exciting electron must remove
an atomic electron from an inner shell. The resulting hole is
filled by outer electrons, and their binding energy is
released in the form of characteristic light quanta. All
transitions which end on the same inner shell occur together
and form a series.
Schematic explanation of the K, L and M series in x-ray spectra. Left: an electron
hole is formed by ionisation of an inner shell. This is filled by an electron from a
shell which is farther out. The binding energy is emitted as an x-ray quantum.
Right: the same in the form of a term scheme. The ionisation limit is shaded in at
the top.
The quantitative observations of the relations between the
frequency and the nuclear charge thus become understandable:
the atomic number Z is screened by one elementary charge in
the K shell and by 7.4 e in the L shell for the electron making
the transition.
The transitions involving inner shells are much more energetic
than those in the outermost shell, because the nuclear charge is
shielded only by those electrons in still lower shells. This
results in screening to a charge (Z-1) for the K lines, and to
(Z-7.4) for the L lines. The field strength in the interior of a
sphere with a uniformly charged surface is zero, so the
external electrons make no contribution to the field
experienced by the inner ones.
Fine structure of the x-ray spectra
The x-ray transitions indicated by Greek letters, K, K, L, L,
start from terms with different principal quantum numbers n.
The fine structure of the x-ray spectra is the occurrence of
several components in a given transition, because of the
splitting of the energy terms resulting from the spin-orbit
coupling of the electrons in inner shells.
Similarly to the spectra of alkali atoms, the x-ray spectra can
be understood as one-electron (or one-hole) spectra. A missing
electron (or a hole) in a full shell is equivalent to a single
electron in an empty shell.
Fine structure diagram
for the x-ray spectra of a
platinum anode
Optical selection rule:
l = 1, j = 0, 1
X-ray absorption
scattering
Incident x-ray I0
Transmitted x-ray I
x
I  I 0 exp( x)
Where µ is the extinction coefficient, which is the sum of
scattering and absorption.
X-ray absorption spectra
X-ray absorption spectra:
the dependence of the absorption coefficient on quantum energy,
i.e., the spectral distribution in absorption spectra.
X-ray absorption spectra typically display a large decrease,
absorption edges, in the absorption coefficient with increasing
quantum number, which are quantum energies at which the
absorption coefficient jumps to a higher value.
These edges correspond to the series limits for the K, L, M, …
series. The subshells also appear as edges, for example LI, LII,
and LIII.
In order for an atom to absorb x-radiation, an electron must
be excited from an inner shell into a less strongly bound state.
Since the neighboring shells are already occupied, discrete
absorption lines due to transitions from one shell into another
are scarcely observable. There is, however, a continuum of
free states on the other side of the series limit into which the
absorption spectra are the superimposed seires limit continua
of the various shells and the subshells.
Absorption edges are located at those points where the
energy of the x-ray quantum is just sufficient to allow an
absorptive transition from a new shell into the limiting
continuum.
At lower frequencies, the quantum energy h is only sufficient
to release electrons from outer shells. As h increases, and an
energy is reached which is sufficient to release even K electrons,
and at this point the absorption coefficient increases abruptly.
The fine structure of the absorption edges is further evidence for
the existence of shells and subshells: there is one K edge, but 3L
edges, 5M edges, and so on.
If the spectral resolution is good enough, it is possible to detect
effects of chemical bonding on the energies and fine structure of
the absorption edges.
Aside from the edges, the frequency dependence of the
absorption coefficient is essentially expressed by:
 abs  3 Z x
with
3 x  4
The hardness or penetrating ability of the x-rays thus increases
as the wavelength decreases.
The Auger effect
The Auger effect was discovered by P. Auger in 1923.
After an electron has been removed from an inner shell, the
excess energy can be released either in the form of an x-ray
quantum, or non-radiative return to the ground state with the
emission of an electron from a shell farther out (Auger effect).
The non-radiative processes competes with x-ray emission. The
observed quantum yield for x-ray emission  is less than 1:
 = number of x-ray emitting atoms/number of ionised atoms
 increases with increasing nuclear charge. In light atoms, the
non-radiative processes far outweigh emittive processes.
The quantum yield for the emission of x-rays as a
function of the Z number
Auger electron emission competes with x-ray emission
The kinetic energy of the Auger electron:
Ekin  h K  EL  ( EK  EL )  EL
A numerical example: Ag is bombarded with K radiation
from a W anticathode (59.1 keV). Electrons with the
following energies are observed:
1) 55.8 keV: Photoelectrons from the Ag L shell
The ionisation energy of the Ag L shell EionL = 3.34 keV
Ekin = 59.1 – 3.34 = 55.76 keV
2) 33.8 keV: photoelectrons from the Ag K shell
EionK = 25.4 keV, Ekin = 59.1 – 25.4 = 33.7 keV
3) 21.3 keV: Auger electrons
EK(Ag) – EionL = 24.9 – 3.34 = 21.56 keV
4) 18.6 keV: Auger electrons
EK(Ag) – EionL = 22.1 – 3.34 = 18.76 keV
Ag
M
ESCA (Electron Spectroscopy for Chemical Analysis)
ESCA was developed in particular by K. Siegbahn and
coworkers. It has become an important experimental technique
in chemistry and in molecular and solid state physics.
Since the binding energies of the electrons are characteristic of
the particular atoms, the measurements of the kinetic energy of
the emitted electrons (photoelectrons or Auger electrons) can be
used for chemical analysis of a sample. Furthermore, the
chemical bonding between atoms in molecules or in solids leads
to a redistribution of the valence electrons. The resulting small
shifts (chemical shifts) can also be measured.
XPS: x-ray photoelectron spectroscopy
AES: Auger electron spectroscopy
Sources: x-ray, UV light, electron beam, synchrotron radiation
XPS for SnO:Co
Sn 3d3/2
Sn 3d5/2
Sn MNN
O KLL
Sn 3p3/2
Sn 3p1/2
Co 3d1/2
Co 3d3/2
O1s
homework
pp322
18.1, 18.5, 18.6